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If $R : H \to \mathbb{R}$ be a functional with $H$ be a Hilbert space. I want to how convexity of $R$ (If it is convex) is related with weak lower semi continuity??

By definition weak lower semi continuity : Let $(x_n)_n$ a sequence such that $x_n \to x$ in $H$, then $x_n \to x$ weakly in $H$, then $$R(x) \leq \liminf_{n \to \infty} R(x_n).$$ where topology on $H$ is weak topology.

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Convexity plus (strong) lower semicontinuity implies weakly lower semicontinuity.

Just argue by using epigraphs: If $R$ is convex and strong lower semicontinuous, then its epigraph is convex and closed, hence weakly closed, hence $R$ is weakly lower semicontinuous.

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  • $\begingroup$ Do you have a reference to this result? I may need to site it but I don;t have space to prove it :) $\endgroup$ – AIM_BLB Jun 20 '18 at 13:05

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